Average Error: 14.0 → 8.4
Time: 27.3s
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{1}{\frac{\frac{\ell}{{\left(\frac{\sqrt[3]{M} \cdot \sqrt[3]{M}}{\frac{d}{D} \cdot \frac{2}{\sqrt[3]{M}}}\right)}^{\left(\frac{2}{2}\right)}}}{{\left(\frac{M}{\frac{d}{D} \cdot 2}\right)}^{\left(\frac{2}{2}\right)} \cdot h}}}\]
w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}
↓
w0 \cdot \sqrt{1 - \frac{1}{\frac{\frac{\ell}{{\left(\frac{\sqrt[3]{M} \cdot \sqrt[3]{M}}{\frac{d}{D} \cdot \frac{2}{\sqrt[3]{M}}}\right)}^{\left(\frac{2}{2}\right)}}}{{\left(\frac{M}{\frac{d}{D} \cdot 2}\right)}^{\left(\frac{2}{2}\right)} \cdot h}}}
double f(double w0, double M, double D, double h, double l, double d) {
        double r144827 = w0;
        double r144828 = 1.0;
        double r144829 = M;
        double r144830 = D;
        double r144831 = r144829 * r144830;
        double r144832 = 2.0;
        double r144833 = d;
        double r144834 = r144832 * r144833;
        double r144835 = r144831 / r144834;
        double r144836 = pow(r144835, r144832);
        double r144837 = h;
        double r144838 = l;
        double r144839 = r144837 / r144838;
        double r144840 = r144836 * r144839;
        double r144841 = r144828 - r144840;
        double r144842 = sqrt(r144841);
        double r144843 = r144827 * r144842;
        return r144843;
}
↓
double f(double w0, double M, double D, double h, double l, double d) {
        double r144844 = w0;
        double r144845 = 1.0;
        double r144846 = 1.0;
        double r144847 = l;
        double r144848 = M;
        double r144849 = cbrt(r144848);
        double r144850 = r144849 * r144849;
        double r144851 = d;
        double r144852 = D;
        double r144853 = r144851 / r144852;
        double r144854 = 2.0;
        double r144855 = r144854 / r144849;
        double r144856 = r144853 * r144855;
        double r144857 = r144850 / r144856;
        double r144858 = 2.0;
        double r144859 = r144854 / r144858;
        double r144860 = pow(r144857, r144859);
        double r144861 = r144847 / r144860;
        double r144862 = r144853 * r144854;
        double r144863 = r144848 / r144862;
        double r144864 = pow(r144863, r144859);
        double r144865 = h;
        double r144866 = r144864 * r144865;
        double r144867 = r144861 / r144866;
        double r144868 = r144846 / r144867;
        double r144869 = r144845 - r144868;
        double r144870 = sqrt(r144869);
        double r144871 = r144844 * r144870;
        return r144871;
}

Error

Bits error versus w0

Bits error versus M

Bits error versus D

Bits error versus h

Bits error versus l

Bits error versus d

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Initial program 14.0

    \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
  2. Using strategy rm

  3. Applied associate-*r/10.8

    \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot h}{\ell}}}\]

  4. Simplified10.7

    \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{{\left(\frac{M}{\frac{2}{D} \cdot d}\right)}^{2} \cdot h}}{\ell}}\]
  5. Using strategy rm

  6. Applied sqr-pow10.7

    \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{\left({\left(\frac{M}{\frac{2}{D} \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{M}{\frac{2}{D} \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot h}{\ell}}\]

  7. Applied associate-*l*9.1

    \[\leadsto w0 \cdot \sqrt{1 - \frac{\color{blue}{{\left(\frac{M}{\frac{2}{D} \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left({\left(\frac{M}{\frac{2}{D} \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot h\right)}}{\ell}}\]

  8. Simplified9.1

    \[\leadsto w0 \cdot \sqrt{1 - \frac{{\left(\frac{M}{\frac{2}{D} \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \color{blue}{\left(h \cdot {\left(\frac{\frac{M}{2}}{\frac{d}{D}}\right)}^{\left(\frac{2}{2}\right)}\right)}}{\ell}}\]
  9. Using strategy rm

  10. Applied clear-num9.1

    \[\leadsto w0 \cdot \sqrt{1 - \color{blue}{\frac{1}{\frac{\ell}{{\left(\frac{M}{\frac{2}{D} \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(h \cdot {\left(\frac{\frac{M}{2}}{\frac{d}{D}}\right)}^{\left(\frac{2}{2}\right)}\right)}}}}\]

  11. Simplified8.4

    \[\leadsto w0 \cdot \sqrt{1 - \frac{1}{\color{blue}{\frac{\frac{\ell}{{\left(\frac{M}{2 \cdot \frac{d}{D}}\right)}^{\left(\frac{2}{2}\right)}}}{{\left(\frac{M}{2 \cdot \frac{d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot h}}}}\]
  12. Using strategy rm

  13. Applied add-cube-cbrt8.4

    \[\leadsto w0 \cdot \sqrt{1 - \frac{1}{\frac{\frac{\ell}{{\left(\frac{\color{blue}{\left(\sqrt[3]{M} \cdot \sqrt[3]{M}\right) \cdot \sqrt[3]{M}}}{2 \cdot \frac{d}{D}}\right)}^{\left(\frac{2}{2}\right)}}}{{\left(\frac{M}{2 \cdot \frac{d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot h}}}\]

  14. Applied associate-/l*8.4

    \[\leadsto w0 \cdot \sqrt{1 - \frac{1}{\frac{\frac{\ell}{{\color{blue}{\left(\frac{\sqrt[3]{M} \cdot \sqrt[3]{M}}{\frac{2 \cdot \frac{d}{D}}{\sqrt[3]{M}}}\right)}}^{\left(\frac{2}{2}\right)}}}{{\left(\frac{M}{2 \cdot \frac{d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot h}}}\]

  15. Simplified8.4

    \[\leadsto w0 \cdot \sqrt{1 - \frac{1}{\frac{\frac{\ell}{{\left(\frac{\sqrt[3]{M} \cdot \sqrt[3]{M}}{\color{blue}{\frac{2}{\sqrt[3]{M}} \cdot \frac{d}{D}}}\right)}^{\left(\frac{2}{2}\right)}}}{{\left(\frac{M}{2 \cdot \frac{d}{D}}\right)}^{\left(\frac{2}{2}\right)} \cdot h}}}\]

  16. Final simplification8.4

    \[\leadsto w0 \cdot \sqrt{1 - \frac{1}{\frac{\frac{\ell}{{\left(\frac{\sqrt[3]{M} \cdot \sqrt[3]{M}}{\frac{d}{D} \cdot \frac{2}{\sqrt[3]{M}}}\right)}^{\left(\frac{2}{2}\right)}}}{{\left(\frac{M}{\frac{d}{D} \cdot 2}\right)}^{\left(\frac{2}{2}\right)} \cdot h}}}\]

Reproduce

herbie shell --seed 2019196 +o rules:numerics
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))